Problem: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{r^2 - 25}{r - 5}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{25} = -5$ So we can rewrite the expression as: $k = \dfrac{({r} {-5})({r} + {5})} {r - 5} $ We can divide the numerator and denominator by $(r - 5)$ on condition that $r \neq 5$ Therefore $k = r + 5; r \neq 5$